Automorphisms of order 3 — the known graphs recovered, the open parameters mapped

Method, in one paragraph. Let σ be an automorphism of order 3 of an srg(n,k,λ,μ) with integer eigenvalues r > s, d = r − s, f fixed vertices and o = (n−f)/3 orbits. The adjacency matrix restricted to a nontrivial character of ⟨σ⟩ is an o×o Hermitian block C over Z[ω] with C_ii ∈ {0, −1} and off-diagonal entries 0 or units, and (C − rI)(C − sI) = 0. Two consequences are checkable without any orbit matrix: (i) with G = rI − C, G² = dG forces every row to contain exactly g_ii(d − g_ii) unit entries — r(d−r) if a_ii = 0, (r+1)(d−r−1) if a_ii = 2 — so a row type needs o − 1 ≥ that many columns; (ii) the trace of C equals −n₂ (n₂ = number of orbits with a_ii = 2), so r·m + s·(o−m) = −n₂ must have an integer solution m (the r-multiplicity of the block). When no (n₂, m) survives, the fixed-point count f is impossible (IMPOSSIBLE). When one survives with a small rank a = min(m, o−m), the existence of the block is decided exactly by a CP-SAT model over Z[ω] tight frames (tools/eisenlift/frame_enum_cpsat2.py): every basis-Gram class INFEASIBLE means CLOSED. The first table shows the same pipeline on known graphs, where it must (and does) find what exists; the second shows the open parameter sets.